SECTION
3.4
Operations on Functions and Composition of Functions
SKILLS OBJECTIVES
■  
 Add, subtract, multiply, and divide functions.
■  
 Evaluate composite functions.
■  
 Determine the domain of functions resulting from operations on and composition of functions.
CONCEPTUAL OBJECTIVES
■  
 Understand domain restrictions when dividing functions.
■  
 Realize that the domain of a composition of functions excludes values that are not in the domain of the inside function.
 
Two different functions can be combined using mathematical operations such as addition, subtraction, multiplication, and division. Also, there is an operation on functions called composition, which can be thought of as a function of a function. When we combine functions, we do so algebraically. Special attention must be paid to the domain and range of the combined functions.
Adding, Subtracting, Multiplying, and Dividing Functions
Consider the two functions and . The domain of both of these functions is the set of all real numbers. Therefore, we can add, subtract, or multiply these functions for any real number .
The result is in fact a new function, which we denote:
The result is in fact a new function, which we denote:
The result is in fact a new function, which we denote:
Although both and are defined for all real numbers , we must restrict so that to form the quotient .
The result is in fact a new function, which we denote:
Two functions can be added, subtracted, and multiplied. The resulting function domain is therefore the intersection of the domains of the two functions. However, for division, any value of (input) that makes the denominator equal to zero must be eliminated from the domain.
The previous examples involved polynomials. The domain of any polynomial is the set of all real numbers. Adding, subtracting, and multiplying polynomials result in other polynomials, which have domains of all real numbers. Let's now investigate operations applied to functions that have a restricted domain.
The domain of the sum function, difference function, or product function is the intersection of the individual domains of the two functions. The quotient function has a similar domain in that it is the intersection of the two domains. However, any values that make the denominator zero must also be eliminated.
Function
Notation
Domain
Sum
Difference
Product
Quotient
We can think of this in the following way: Any number that is in the domain of both the functions is in the domain of the combined function. The exception to this is the quotient function, which also eliminates values that make the denominator equal to zero.
EXAMPLE 1  Operations on Functions: Determining Domains of New Functions
For the functions and , determine the sum function, difference function, product function, and quotient function. State the domain of these four new functions.
Solution
Sum function:
Difference function:
Product function:
Quotient function:
The domain of the square root function is determined by setting the argument under the radical greater than or equal to zero.
The domain of the sum, difference, and product functions is
The quotient function has the additional constraint that the denominator cannot be zero. This implies that , so the domain of the quotient function is .
■
YOUR TURN 
Given the function and , find and state its domain.
 EXAMPLE 2  Quotient Function and Domain Restrictions
Given the functions and , find the quotient function, , and state its domain.
Solution
The quotient function is written as
Domain of
Domain of
The real numbers that are in both the domain for and the domain for are represented by the intersection . Also, the denominator of the quotient function is equal to zero when , so we must eliminate this value from the domain.
■
YOUR TURN 
For the functions given in Example 2, determine the quotient function , and state its domain.
Composition of Functions
Recall that a function maps every element in the domain to exactly one corresponding element in the range as shown in the figure below.
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Suppose there is a sales rack of clothes in a department store. Let correspond to the original price of each item on the rack. These clothes have recently been marked down . Therefore, the function represents the current sale price of each item. You have been invited to a special sale that lets you take off the current sale price and an additional off every item at checkout. The function determines the checkout price. Note that the output of the function is the input of the function as shown in the figure below.
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This is an example of a composition of functions, when the output of one function is the input of another function. It is commonly referred to as a function of a function.
An algebraic example of this is the function . Suppose we let and . Recall that the independent variable in function notation is a placeholder. Since , then . Substituting the expression for , we find . The function is said to be a composite function, .
Note that the domain of is the set of all real numbers, and the domain of is the set of all nonnegative numbers. The domain of a composite function is the set of all such that is in the domain of . For instance, in the composite function , we know that the allowable inputs into are all numbers greater than or equal to zero. Therefore, we restrict the outputs of and find the corresponding . Those are the only allowable inputs and constitute the domain of the composite function .
The symbol that represents composition of functions is a small open circle; thus and is read aloud as “ of .” It is important not to confuse this with the multiplication sign: .
CAUTION 
COMPOSITION OF FUNCTIONS
Given two functions and , there are two composite functions that can be formed.
Notation
Words
Definition
Domain
composed with
The set of all real numbers in the domain of such that is also in the domain of .
composed with
The set of all real numbers in the domain of such that is also in the domain of .
It is important to realize that there are two “filters” that allow certain values of into the domain. The first filter is . If is not in the domain of , it cannot be in the domain of . Of those values for that are in the domain of , only some pass through, because we restrict the output of to values that are allowable as input into . This adds an additional filter.
The domain of is always a subset of the domain of , and the range of is always a subset of the range of .
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 EXAMPLE 3  Finding a Composite Function
Given the functions and , find .
Solution
Write using placeholder notation.
Express the composite function .
Substitute into .
Eliminate the parentheses on the right side.
■
YOUR TURN 
Given the functions in Example 3, find .
 EXAMPLE 4  Determining the Domain of a Composite Function
Given the functions and , determine , and state its domain.
Solution
Write using placeholder notation.
Express the composite function .
Substitute into .
Multiply the right side by .
What is the domain of ? By inspecting the final result of , we see that the denominator is zero when . Therefore, . Are there any other values for that are not allowed? The function has the domain ; therefore we must also exclude zero.
The domain of excludes and or, in interval notation,
■
YOUR TURN 
For the functions and given in Example 4, determine the composite function and state its domain.
The domain of the composite function cannot always be determined by examining the final form of .
CAUTION 
The domain of the composite function cannot always be determined by examining the final form of .
EXAMPLE 5  Determining the Domain of a Composite Function (Without Finding the Composite Function)
Let and . Find the domain of . Do not find the composite function.
Solution
Find the domain of .
Find the range of .
In , the output of becomes the input for . Since the domain of is the set of all real numbers except 2, we eliminate any values of in the domain of that correspond to .
Let .
Square both sides.
Solve for .
Eliminate from the domain of , .
State the domain of .
 EXAMPLE 6  Evaluating a Composite Function
Given the functions and , evaluate:
(a)  
(b)  
(c)  
(d)  
Solution
One way of evaluating these composite functions is to calculate the two individual composites in terms of : and . Once those functions are known, the values can be substituted for and evaluated.
Another way of proceeding is as follows:
(a)  
Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .
(b)  
Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .
(c)  
Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .
(d)  
Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .
■
YOUR TURN 
Given the functions and , evaluate and .
3.4.2.1 Application Problems
Recall the example at the beginning of this chapter regarding the clothes that are on sale. Often, real-world applications are modeled with composite functions. In the clothes example, is the original price of each item. The first function maps its input (original price) to an output (sale price). The second function maps its input (sale price) to an output (checkout price). Example 7 is another real-world application of composite functions.
Three temperature scales are commonly used:
■  
 The degree Celsius scale
●  
 This scale was devised by dividing the range between the freezing and boiling points of pure water at sea level into 100 equal parts. This scale is used in science and is one of the standards of the “metric” (SI) system of measurements.
■  
 The Kelvin (K) temperature scale
●  
 This scale shifts the Celsius scale down so that the zero point is equal to absolute zero (about ), a hypothetical temperature at which there is a complete absence of heat energy.
●  
 Temperatures on this scale are called kelvins, not degrees kelvin, and kelvin is not capitalized. The symbol for the kelvin is K.
■  
 The degree Fahrenheit scale
●  
 This scale evolved over time and is still widely used mainly in the United States, although Celsius is the preferred “metric” scale.
●  
 With respect to pure water at sea level, the degrees Fahrenheit are gauged by the spread from (freezing) to (boiling).
The equations that relate these temperature scales are
EXAMPLE 7  Applications Involving Composite Functions
Determine degrees Fahrenheit as a function of kelvins.
Solution
Degrees Fahrenheit is a function of degrees Celsius.
Now substitute into the equation for .
Simplify.
 
  SECTION
3.4 		SUMMARY
Operations on Functions
Function
Notation
Sum
Difference
Product
Quotient
The domain of the sum, difference, and product functions is the intersection of the domains, or common domain shared by both and . The domain of the quotient function is also the intersection of the domain shared by both and with an additional restriction that .
Composition of Functions
The domain restrictions cannot always be determined simply by inspecting the final form of . Rather, the domain of the composite function is a subset of the domain of . Values of must be eliminated if their corresponding values of are not in the domain of .
SECTION
3.4 		EXERCISES
SKILLS
In Exercises 1-10, given the functions and , find , , , and , and state the domain of each.
1.  
2.  
3.  
4.  
5.  
6.  
7.  
8.  
9.  
10.  
In Exercises 11-20, for the given functions and , find the composite functions and , and state their domains.
11.  
12.  
13.  
14.  
15.  
16.  
17.  
18.  
19.  
20.  
In Exercises 21-38, evaluate the functions for the specified values, if possible.
21.  
22.  
23.  
24.  
25.  
26.  
27.  
28.  
29.  
30.  
31.  
32.  
33.  
34.  
35.  
36.  
37.  
38.  
In Exercises 39-50, evaluate and , if possible.
39.  
40.  
41.  
42.  
43.  
44.  
45.  
46.  
47.  
48.  
49.  
50.  
In Exercises 51-60, show that and .
51.  
52.  
53.  
54.  
55.  
56.  
57.  
58.  
59.  
60.  
In Exercises 61-66, write the function as a composite of two functions and . (More than one answer is correct.)
61.  
62.  
63.  
64.  
65.  
66.  
APPLICATIONS
Exercises 67 and 68 depend on the relationship between degrees Fahrenheit, degrees Celsius, and kelvins:
67.  
Temperature.
Write a composite function that converts kelvins into degrees Fahrenheit.
68.  
Temperature.
Convert the following degrees Fahrenheit to kelvins: and .
69.  
Dog Run.
Suppose that you want to build a square fenced-in area for your dog. Fencing is purchased in linear feet.
(a)  
Write a composite function that determines the area of your dog pen as a function of how many linear feet are purchased.
(b)  
If you purchase 100 linear feet, what is the area of your dog pen?
(c)  
If you purchase 200 linear feet, what is the area of your dog pen?
70.  
Dog Run.
Suppose that you want to build a circular fenced-in area for your dog. Fencing is purchased in linear feet.
(a)  
Write a composite function that determines the area of your dog pen as a function of how many linear feet are purchased.
(b)  
If you purchase 100 linear feet, what is the area of your dog pen?
(c)  
If you purchase 200 linear feet, what is the area of your dog pen?
71.  
Market Price.
Typical supply and demand relationships state that as the number of units for sale increases, the market price decreases. Assume that the market price and the number of units for sale are related by the demand equation:
Assume that the cost of producing items is governed by the equation
and the revenue generated by selling units is governed by
(a)  
Write the cost as a function of price .
(b)  
Write the revenue as a function of price .
(c)  
Write the profit as a function of price .
72.  
Market Price.
Typical supply and demand relationships state that as the number of units for sale increases, the market price decreases. Assume that the market price and the number of units for sale are related by the demand equation:
Assume that the cost of producing items is governed by the equation
and the revenue generated by selling units is governed by
(a)  
Write the cost as a function of price .
(b)  
Write the revenue as a function of price .
(c)  
Write the profit as a function of price .
In Exercises 73 and 74, refer to the following:
The cost of manufacturing a product is a function of the number of hours the assembly line is running per day. The number of products manufactured is a function of the number of hours the assembly line is operating and is given by the function . The cost of manufacturing the product measured in thousands of dollars is a function of the quantity manufactured, that is, the function .
73.  
Business.
If the quantity of a product manufactured during a day is given by
and the cost of manufacturing the product is given by
(a)  
Find a function that gives the cost of manufacturing the product in terms of the number of hours the assembly line was functioning, .
(b)  
Find the cost of production on a day when the assembly line was running for 16 hours. Interpret your answer.
74.  
Business.
If the quantity of a product manufactured during a day is given by
and the cost of manufacturing the product is given by
(a)  
Find a function that gives the cost of manufacturing the product in terms of the number of hours the assembly line was functioning, .
(b)  
Find the cost of production on a day when the assembly line was running for 24 hours. Interpret your answer.
In Exercises 75 and 76, refer to the following:
Surveys performed immediately following an accidental oil spill at sea indicate the oil moved outward from the source of the spill in a nearly circular pattern. The radius of the oil spill measured in miles is a function of time measured in days from the start of the spill, while the area of the oil spill is a function of radius, that is, the function .
75.  
Environment: Oil Spill.
If the radius of the oil spill is given by
and the area of the oil spill is given by
(a)  
Find a function that gives the area of the oil spill in terms of the number of days since the start of the spill, .
(b)  
Find the area of the oil spill to the nearest square mile 7 days after the start of the spill.
76.  
Environment: Oil Spill.
If the radius of the oil spill is given by
and the area of the oil spill is given by
(a)  
Find a function that gives the area of the oil spill in terms of the number of days since the start of the spill, .
(b)  
Find the area of the oil spill to the nearest square mile 5 days after the start of the spill.
77.  
Environment: Oil Spill.
An oil spill makes a circular pattern around a ship such that the radius in feet grows as a function of time in hours . Find the area of the spill as a function of time.
78.  
Pool Volume.
A 20 foot by 10 foot rectangular pool has been built. If 50 cubic feet of water is pumped into the pool per hour, write the water-level height (feet) as a function of time (hours).
79.  
Fireworks.
A family is watching a fireworks display. If the family is 2 miles from where the fireworks are being launched and the fireworks travel vertically, what is the distance between the family and the fireworks as a function of height above ground?
80.  
Real Estate.
A couple are about to put their house up for sale. They bought the house for a few years ago, and when they list it with a realtor they will pay a commission. Write a function that represents the amount of money they will make on their home as a function of the asking price .
CATCH THE MISTAKE
In Exercises 81-86, for the functions and , find the indicated function and state its domain. Explain the mistake that is made in each problem.
81.  
Solution:
Domain:
This is incorrect. What mistake was made?
82.  
Solution:
Domain:
This is incorrect. What mistake was made?
83.  
Solution:
Domain:
This is incorrect. What mistake was made?
84.  
Given the function and , find , and state the domain.
Solution:
Domain:
This is incorrect. What mistake was made?
85.  
Domain:
This is incorrect. What mistake was made?
86.  
Domain:
This is incorrect. What mistake was made?
CONCEPTUAL
In Exercises 87-90, determine whether each statement is true or false.
87.  
When adding, subtracting, multiplying, or dividing two functions, the domain of the resulting function is the union of the domains of the individual functions.
88.  
For any functions and , for all values of that are in the domain of both and .
89.  
For any functions and , exists for all values of that are in the domain of , provided the range of is a subset of the domain of .
90.  
The domain of a composite function can be found by inspection, without knowledge of the domain of the individual functions.
CHALLENGE
91.  
For the functions and , find and state its domain.
92.  
For the functions and , find and state its domain.
93.  
For the functions and find and state its domain.
94.  
For the functions and , find and state its domain. Assume and .
TECHNOLOGY
95.  
Using a graphing utility, plot and . Plot . What is the domain of ?
96.  
Using a graphing utility, plot , , and . What is the domain of ?
97.  
Using a graphing utility, plot , , and . If represents a function and represents a function , then represents the composite function . The graph of is only defined for the domain of . State the domain of .
98.  
Using a graphing utility, plot , and . If represents a function and represents a function , then represents the composite function . The graph of is only defined for the domain of . State the domain of .


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